Review211Final - Nonlinear Programming Review Nonlinear...

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MS&E 211 Nonlinear Review Nonlinear Programming Review Nonlinear Programming Types of Nonlinear Programs (NLP) Convexity and Convex Programs NLP Solutions Unconstrained Optimization Principles of Unconstrained Optimization Search Methods Constrained Optimization Theory The KKT Conditions The Lagrange Multiplier (Sensitivity Analysis) Linearly Constrained Optimization (LCP) Duality and optimality conditions revisited Solution concepts for Quadratic Programs (QP) and LCP Classification of NLP Algorithms and Solution Methods
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MS&E 211 Nonlinear Review Nonlinear Optimization Model 0 ) ( 0 ) ( 0 ) ( ) ( 2 1 m 2 1 2 2 1 1 2 1 s.t. min x x x c x x x c x x x c x x x f n n n n ,..., , ,..., , ,..., , ,..., ,
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MS&E 211 Nonlinear Review Types of Nonlinear Programs Unconstrained optimization. Linearly constrained optimization Quadratic Programming Convex optimization Objective function is a convex function in minimization or a concave function in maximization (over the feasible set) Feasible set is a convex set Nonconvex optimization Geometric programming Fractional programming
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MS&E 211 Nonlinear Review Gradient Vector and Hessian Matrix The gradient vector of f at x = n 2 1 ... ) ( x f x f x f x f = n n 2 n 1 2 1 n 2 1 1 2 2 ... ... ... ... ... ) ( x x f x x f x x f x x f x f The Hessian Matrix of f at x
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MS&E 211 Nonlinear Review Convex and Concave Functions f(x) f(x) x x f(x) is a convex function if and only if for any given two points x 1 and x 2 in the function domain and for any constant 0 α 1 f( α x 1 +(1- α )x 2 ) α f(x 1 )+(1- α )f(x 2 ) x 1 x 2 f(x 1 ) f(x 2 ) α x 1 +(1- α )x 2 α f(x 1 )+(1- α )f(x 2 ) f( α x 1 +(1- α )x 2 )
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MS&E 211 Nonlinear Review Properties of Convex Function b x f(x) If f(x) is a convex function, then the lower level set { x: f(x) b } is a convex set for any constant b . The graph of a convex function lies above its tangent planes. The Hessian matrix of a convex function is positive semi-definite.
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MS&E 211 Nonlinear Review Convex Quadratic Function f(x)=x T Qx+c T x is a convex function if and only if Q is positive semidefinite. f(x)=x T Qx+c T x is a strictly convex function if and only if Q is positive definite. If Q is positive definite, Q -1 exists.
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MS&E 211 Nonlinear Review Convex Sets A set is convex if every line segment connecting any two points in the set is contained entirely within the set Ex - polyhedron Ex - ball An extreme point of a convex set is any point that is not on any line segment connecting any other two points of the set The intersection of convex sets is a convex set
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MS&E 211 Nonlinear Review Why do we care so much about convexity?
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